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Heavy top quark and scale dependence of quark mixing
Volume 257, number 3,4
PHYSICS LETTERS B
28 March 1991
Heavy top quark and scale dependence of quark mixing Marek Olechowski Max-Planck-lnstitut J~r Physik und Astrophysik, Werner-tIeisenberg-lnstitut ~r Physik, P.O. Box 40 12 12, W-8000 Munich, FRG
and Stephan Pokorski Centre de Physique Th~orique, [2cole Polytechnique, F-91128 Palaiseau, France and Institute for Theoretical Physics, Warsaw University, ul. Hoffa 69, PL-O0-681 Warsaw, Poland
Received 9 November 1990; revised manuscript received 11 January 1991
One-loop renormalization group equations for the quark mixing elements [ Vul and for the ratios h~/hj of the eigenvalues of the Yukawa coupling matrices are derived in analytic form for a large class of theories and solved in the presence of large third generation Yukawa couplings. Comparison of scale dependence of the elements [ ~j[ and of the ratios h,/hj is also given. The possible relevance of the evolution effects for understanding the pattern of quark masses and mixing is discussed. The present experimental limit on the top quark mass encourages us to discuss the scale d e p e n d e n c e o f the other quark mass parameters, such as the elements I V01 of the mixing matrix and the ratios of the quark masses. The actual value o f the top mass itself may even be related to the structure o f the renormalization group ( R G ) equation for Yukawa couplings, namely its infra-red ( I R ) fixed point [ 1 ] (this idea has recently been d e v e l o p e d in a u m b e r o f specific scenarios [ 2 ] ) . Although the low energy physical values of the other quark mass and mixing p a r a m e ters are not the infra-red fixed point values (this would imply, among other things, the vanishing o f the quark m i x i n g ) , scale d e p e n d e n c e o f those p a r a m eters is no longer negligible in the presence of large Yukawa coupling (s) o f the third generation quarks. There are at least two obvious reasons for investigating this scale dependence, both related to the p r o b l e m o f u n d e r s t a n d i n g the pattern o f fermion masses. Firstly, given the experimental values o f the physical p a r a m e t e r s at the scale Mw, it is i m p o r t a n t to know their "initial" values at the, presumably much On leave of absence from Institute for Theoretical Physics, Warsaw University, ul. Ho~,a69, PL-00-681 Warsaw, Poland. 388
higher, scale where the structure o f the Yukawa coupling matrices is d e t e r m i n e d by, as yet unknown, physics. Secondly, in search for this new physics principles, there have been m a n y phenomenological attempts to relate the quark mixing p a r a m e t e r s to the values of the quark masses. It is an almost obvious necessary physical requirement that any such p a r a m etrization, to be considered as the one which m a y indeed reflect some underlying physics at some higher energy scale, should be stable against the R G evolution up to that scale. Therefore, it is interesting to c o m p a r e the scale dependence o f the elements I Vu[ with the scale dependence of various ratios hi~hi o f the eigenvalues o f Yukawa coupling matrices. Scale dependence is usually studied by means o f R G equations and the one-loop R G equations for the Yukawa matrices are well known in a large class o f theories [3]. However, to study the scale dependence of the observables I V,;[ and mi/rnj it is convenient ( a n d for several purposes even necessary) to have the R G equations for those observables in analytic form. In this paper we derive the R G equations for the mixing elements I V0[ and for the ratios h~/hj o f the eigenvalues of Yukawa coupling matrices. This is
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
Volume 257, number 3,4
PHYSICS LETTERS B
possible by use of the recently proposed [4 ] weak basis invariants directly related to the elements I V,jl. Next, we study the scale dependence of those quantities up to the unification scale in the presence of large Yukawa coupling(s) for the third generation quarks. Of course, the scale dependence of those physical parameters depends on the theory considered. After developing the general formalism we focus on the minimal standard model (MSM) and on the minimal SUSY model (SUSY). In this latter case we consider the possibility h~>> hb (and consequently v~~ v2) and also h~~ h b (b'2 > > / h ). W e observe interesting qualitative features of the scale dependence of various parameters and comment on their possible relevance for understanding the pattern of quark masses. (Our discussion for the SUSY model remains valid with only minor modifications for the general twoHiggs doublet models. ) The subject of the scale dependence of the quark mixing parameters has already been studied in the past by several authors [5-7]. In particular in ref. [6] the RG equations are given in analytic form for the elements [ V~j[ in the standard model and in ref. [7] the scale dependence of the mixing angles has been studied (by diagonalizing numerically the Yukawa coupling matrices at different scales) also in two-Higgs doublet models and in SUSY models in the presence of one large Yukawa coupling. We generalize those results in several directions. First, we derive the one-loop RG equations for the mixing elements I V,I in a large class of theories, including the cases studied earlier numerically. Thus, a qualitative insight into the behavior of different mixing angles and of the CP violation measure Joe in various models is obtained, our equations allow also for an easy study of models with two (three) large Yukawa couplings (top and bottom (and tau) couplings). Second, we also derive RG equations for the ratios of the eigenvalues of the Yukawa coupling matrices and compare them at the analytic level with the equations for I Vgjl. This may have interesting implications for models of the quark mass generation, which we briefly discuss. Let us consider the one-loop RG equations for the Yukawa matrices Y4, (A = U, D), in the following general form: Y~ = c,4 ~ + ~ aA~HB YA,
( 1)
28 March 1991
where H,~ = Y4 Y,+ is a hermitian matrix and t = ( 1/ 16n 2) ln(Mx//~). Eq. ( 1 ) includes the three specific cases we consider in this paper: ( 1 ) MSM with one Higgs doublet: CA "= G.4 -
T,
T = 3 Tr Hu + 3 Tr HD + T r H v , GU
Q 2.d_9 2_1_17 2 = o g 3 - - ~ g 2 - - ~-0gt ,
G D = 8 g 3 2 "t-~g2 9 2 "1- ~I g l2 , 3
auu=aDD=--5,
3 aUD-----aDU=~.
(2) Two-Higgs doublet model with one doublet giving mass to the up quarks and the other one to the down quarks and leptons: cA = G A - G ,
Tu=3TrHu,
TD=3TrHD+TrHE,
Gu = 8g 2 + 9g22+ ~2og~, 9 2
1 2
GD = 8g~ +~g2 + ~g~ , auu
=aDD
3 = - - 2,
aUD=aDU=--~
J
.
(3) SUSY model: c4 =GA -- TA,
TU = 3 T r H U , G
TD=3TrHD+TrHE,
v +~g2, u = y16g 32 --v3g5
GD=!~g~ + 3g 2 +~g17 2 , auu=aDD=--3,
aUD=aDU
=-1
,
where HE = YEY~- and YE is the Yukawa matrix for charged leptons. We notice an important difference in the coefficients ofeq. ( 1 ) between the MSM and the other two models: det ( a AB ) -~ 0 for MSM and det (aA t~ ) ~= 0 o t h e r w i s e . It implies different relations among the couplings in the IR fixed point. The MSM is special in that it allows in this point for the mass hierarchy among generations Hu-HDZC'I (in the approximations when the running of gauge couplings is neglected) whereas in the other cases Hu zc HD 3C~ at the fixed point. Of course, the above conditions are not satisfied in neither case but the approach to the fixed point is different in the two cases. As we shall see, this results in a qualitatively different scale dependence 389
Volume 257, number 3,4
PHYSICS LETTERS B
of various quantities in the MSM and in two-doublet models (including SUSY). From eq. ( 1 ) one can derive R G equations for the elements ]Vv[ and for the ratios hJhj. To this end we use the recently proposed [4 ] set of weak basis invariants l(n, m):
Tr([H~,H~]+[H~j,H'~]) l(n, m ) = Tr(H2, ) Tr(H~ )
(2)
The invariants (2) are directly related to the moduli squared of the mixing elements. For instance lira I(pN, q N ) = 2 ] V~h ]2(1 - ] V~, ] 2) ,
N~ow
(3)
where p, q = +_ 1, a = 2 + p , b - 2 + q . The remaining ] Vu[ 2 can be obtained from the unitarity condition. For the four elements IV, t,] in eq. (3) we have therefore
d
dtln
[Nab[ 2
d lim lnl(pN, qN), (4) -- 1_21V~b12dtx~o~
28 March 1991
sure Jc.e can be obtained from (5) using the relation
[8]
4JZ,e=41Vikl2[ Vtl[ 2 ] Vjk [2 I Viii 2 - ( 1 - I vikl 2 - I v, I 2 - I~kl 2 - [V~tl 2 + [ V~I21 ~ z l 2 + I V. 121 ~k12) 2 ,
where i¢j, k ¢ l . For the ratios of the eigenvalues of the Yukawa coupling matrices we get ~42 d
In ~
e`j
----aDD(h 2, - h 2 ) 3
+aDu ~ h~k(lVk, i2--1V~,i2), k--1
d lnhuJ 2 2 -dth..u~ = a u u ( h u , - h u ~ )
1--[Vab[ 2
3
+auD Z h~k(lVikl2--lVjkl2) k=l
and using eq. (1), after some calculation, one gets the following R G equations ~"
d~ln l
i=l ~" 1
h2i( ((~ih- I Vail2)
2 h 2t,
+ G~,~--~ i j~. h2p_h2f +2qaDu i=l ~
"
(8)
In the presence of large third generation Yukawa couplings, eqs. (5) give to a very good approximation the following: d
-d-t l n [Vi, I = - - a D u h t 2 - - a u D h b2,
]
( i j ) = (13), (31), (23), ( 3 2 ) ,
h2ui(((~ai-[V~b[2)
1
(7)
d - - I n [ VI2 [ = --aDuh2c--auDh2s dt
)
_aouh2 [V31 [2__ [ V1312 ]VI2[ 2
where, as before, p, q = + 1, a = 2 + p and b = 2 + q ; he,, is the i-eigenvalue of the matrix YD (and similarly, h~, of the matrix Yu) and
_auDh 2 [ G3 ]2_ 1V31 [2
R,:,,.t~j= Re ( V~, 17,, if, z, V#)
IVy, [2
= I Vat,[2[ Vq I 2+ [ V,v [21Vm]2+ ( 1 - ~5~i)( 1 - 6,',a) X ( 1 - I V a t , I 2 - I V , jI2-IVo, I 2 - [ V v l 2 ) .
d ~ t l n IV21 I = --aDuh2--auDh 2
(6)
d - - In IJcpl = --2aDuh 2 --2aUD h2. dt
(9)
As mentioned earlier, the RG equations for the remaining [Vql follow from (5) and the unitarity relations. The R G equation for the CP violation mea-
The accuracy of this approximation is very good, the differences between the numerical solutions of eqs. (5) and (9) are below 0.5% for all the cases consid-
~ Eq. (5) is the exact one-loop RG equation for IVc,b]2 if hd, < hd~< hd3 and similarly for the up sector.
~2 In the case of the minimal standard model eqs. (5) and (8) agree with the equations given in ref. [6].
390
Volume 257, number 3,4
PHYSICS LETTERS B
ered. In t h e s a m e a p p r o x i m a t i o n , f o r t h e r a t i o s hi/h I we get din
hal, 2 hb = --aDDhb--aDuh?'
d In ~h,, d--I
=
d In ha dt ~
----
i = 1, 2 ,
--auu h2 --aUD h2,
i = 1, 2 ,
2 _ a D u h 2 - - a D v h ~ ] J/32 --aDDhs
(10)
T h e a n a l y s i s o f eqs. ( 9 ) a n d ( 1 0 ) l e a d s us to t h e f o l l o w i n g c o n c l u s i o n s . In all t h e m o d e l s c o n s i d e r e d in t h i s p a p e r t h e scale d e p e n d e n c e o f t h e C a b i b b o a n gle ] G 2 ] is v e r y w e a k ( i t c h a n g e s b y less t h e n 0.1% between Mw and Mx) and qualitatively different from the behaviour of the small mixing elements. The
- f=lv,a[ - - - f=m,/mb
(a,/
......... f = m j m t
/
/
l a t t e r s h o w s t r o n g scale d e p e n d e n c e i n t h e p r e s e n c e o f large Y u k a w a c o u p l i n g s a n d are very s i m i l a r to e a c h o t h e r for all s m a l l e l e m e n t s . So t h e r a t i o s like I Vl3l / I V23l, etc., r e m a i n a l m o s t c o n s t a n t d u r i n g t h e e v o l u t i o n . H o w e v e r , in t h e M S M t h e s m a l l e l e m e n t s bec o m e l a r g e r w i t h i n c r e a s i n g e n e r g y w h e r e a s in S U S Y ( a n d in t h e g e n e r a l t w o - d o u b l e t c a s e ) t h e y b e c o m e s m a l l e r % T h u s , in M S M t h e n e a r e s t n e i g h b o u r m i x -
]2
d l n h~, = - a u u h2 --aUD h2 - - a u D h S~ I V23 I ,- . dt
',~ 15
28 March 1991
~3 All off-diagonal mixing angles vanish at IR fixed points both in the MSM and in two-doublet models. Thus, one could expect that they should decrease with decreasing energy. The opposite behavior of small mixing angles in two-doublet models is caused by the fact that the observed mass hierarchy among generations (which, according to eqs. (9), is crucial in determining the running of the mixing angles ) is allowed at the fixed point in the MSM and not allowed in two-doublet models (see the discussion before eq. (2)). The fact that the two-doublet models with h, ~ h b might approach fixed point behaviour for both quarks of the third generation does not change this behaviour. 10
(b)
~09 ~" 08 0.7
J r
1.0
E. . . . . . . . . . . .
06
..
-
-
[
--
-
0.5
....
0.5
80
120
,
160
200 240 mtop[GeV]
0.4
80
,
,
,
Iv,a]
f=rns/mb f
,
I00
,
rne/ri1
,
,
t
i
,
120
,
,
,
140
,
,
,
,
,
,
,
160 180 mtop[GeV]
09 08 07 0.6
1 0.5
"~',
~
/
- -
,~
-
-
......
80
~,, 9,
r=/vt=l f=m,/mb f = r n . / m t
100
120
i'
140
160 180 mtop[GeV]
Fig. 1. The ratios of ] V]3], ms~roband me~mr at Mx to their values at Mw as a function of the top quark mass for three models: (a) minimal standard model, (b) SUSY with t,2/v~= 2, (c) SUSY with ht=h b. Mx is the unification scale calculated separately for each model from low energy values of the gauge coupling constants. In the cases (a) and (b) the curve for I G3] overlaps almost the one for ms/mb. The Yukawa couplings ht and hb are evolved according to the RG equations for the traces of the matrices HA, obtained from eq. (1), neglecting light quark contributions. 391
Volume 257, number 3,4
PHYSICS LETTERS B
ing at large (unification?) scale m a y be similar for (12) and ( 2 3 ) families and the observed hierarchy I V231/I V~21 can be partially explained by the evolution in the presence o f a heavy top. In SUSY only Cabibbo mixing remains large at large scale and other mixing elements b e c o m e significantly smaller than at the scale Mw. This b e h a v i o u r o f small angles makes the C a b i b b o angle even m o r e exceptional in twoHiggs doublet models than in the MSM. In the presence o f a heavy top (with a mass close to the IR fixed point value) the scale d e p e n d e n c e o f small angles is a few per cent effect between 1 GeV and 100 GeV. The renormalization of small angles between Mw and the unification scale M x depends strongly on the value o f rn~ a n d on the model considered. Some numerical results are presented in fig. 1. In all the models the CP violation p a r a m e t e r Jcp behaves as a p r o d u c t o f two small mixing elements. The results given in fig. 1a a n d l b are essentially equivalent to those o b t a i n e d in ref. [ 7 ] by a different method. Turning now to a c o m p a r i s o n o f the scale dependence o f the elements I V,~I and o f the ratios h~/hj, we firstly observe that eq. ( 5 ) is different from eq. ( 8 ) . Therefore, the R G evolution does not seem to be naturally c o m p a t i b l e with the possibility that there are exact relations between the mixing elements and the quark masses. However, at the level o f the approxim a t i o n ( 9 ) and (10) there are certain regularities. The scale d e p e n d e n c e o f the C a b i b b o angle is similar to the behaviour o f the ratios hd/hs and hu/hc but only numerically. This d e p e n d e n c e is so weak that we d i d not plot it in fig. 1. F o r small elements I V,j l there is quite clearly a resemblance to the b e h a v i o u r of the h~/hj. In the M S M and in two-doublet models (including S U S Y ) with ht >> ho they run as the ratios hdi/hb, i = 1, 2. On the other hand, in the latter models but with ht'~hb (for v2/Vl >> 1) the angles run as x/~o,/hb ( a n d also as x/h~,/ht ). Thus, the R G evolution is c o m p a t i b l e with some a p p r o x i m a t e relation between the mixing elements (in particular small ele-
392
28 March 1991
m e n t s ) and the ratios o f the quark masses. As a final remark we note that the IR fixed p o i n t value o f the top quark mass in the M S M is m t ~ 230 GeV. In the two-doublet m o d e l it changes from m, ~ 180 GeV for ht >> hb and v2 -~ v~ to mt ~ 240 G e V for ht ~ hb ~4. In the SUSY m o d e l the IR fixed point top mass is smaller: mt ~ 150 GeV for ht >> hb ( 1:2~ vl ) and m~ ~ 190 GeV for h~ ~ h b. One of us (S.P.) would like to acknowledge the very kind hospitality o f T . N . Troung in the Centre de Physique Th6orique, l~cole Polytechnique, Palaiseau, where part o f this work was done. This work is partially s u p p o r t e d by the Polish research project CPBP 01.03. ~4 See also ref. [9].
References [ 1] B. Pendleton and G.G. Ross, Phys. Lett. B 98 ( 1981 ) 291; C.T. Hill, Phys. Rev. D 24 ( 1981 ) 691; E.A. Paschos, Z. Phys. C 26 (1984) 235. [2] J. Kubo, K. Sibold and W. Zimmermann, Phys. Lett. B 220 (1989) 191;Nucl. Phys. B 259 (1985) 331; W.J. Marciano, Phys. Rev. Lett. 62 (1989) 2793; W.A. Bardeen, C.T. Hill and M, Lindner, Phys. Rev. D 41 (1990) 1647. [3] T.P. Cheng, E. Eichten and L.F. Li, Phys. Rev. D 9 (1974) 2259. M. Machacek and M. Voughn, Nucl. Phys. B 236 (1984) 221. [4] M. Olechowski and S. Pokorski, P;,ys. Lett. B 231 (1989) 165. [5] E. Ma and S. Pakvasa, Phys. ~,:~. ? ~6 (1979) 43; Phys. Rev. D20 (1979) 2899. [6] K. Sasaki, Z. Phys. C 32 (1986) 149. [7] B. Grz~dkowski and M. Lindner, Phys. Lett. B 193 (1987) 71; B. Grz~tdkowski, M. Lindner and S. Theisen, Phys. Lett. B 198 (1987) 64. [8] C. Hamzaoui, Phys. Rev. Lett. 61 (1988) 35. [9] C.D. Froggatt, I.D. Knowles and R.G. Moorhouse, Glasgow University preprint GUTPA/90/6-1 (1990).
PHYSICS LETTERS B
28 March 1991
Heavy top quark and scale dependence of quark mixing Marek Olechowski Max-Planck-lnstitut J~r Physik und Astrophysik, Werner-tIeisenberg-lnstitut ~r Physik, P.O. Box 40 12 12, W-8000 Munich, FRG
and Stephan Pokorski Centre de Physique Th~orique, [2cole Polytechnique, F-91128 Palaiseau, France and Institute for Theoretical Physics, Warsaw University, ul. Hoffa 69, PL-O0-681 Warsaw, Poland
Received 9 November 1990; revised manuscript received 11 January 1991
One-loop renormalization group equations for the quark mixing elements [ Vul and for the ratios h~/hj of the eigenvalues of the Yukawa coupling matrices are derived in analytic form for a large class of theories and solved in the presence of large third generation Yukawa couplings. Comparison of scale dependence of the elements [ ~j[ and of the ratios h,/hj is also given. The possible relevance of the evolution effects for understanding the pattern of quark masses and mixing is discussed. The present experimental limit on the top quark mass encourages us to discuss the scale d e p e n d e n c e o f the other quark mass parameters, such as the elements I V01 of the mixing matrix and the ratios of the quark masses. The actual value o f the top mass itself may even be related to the structure o f the renormalization group ( R G ) equation for Yukawa couplings, namely its infra-red ( I R ) fixed point [ 1 ] (this idea has recently been d e v e l o p e d in a u m b e r o f specific scenarios [ 2 ] ) . Although the low energy physical values of the other quark mass and mixing p a r a m e ters are not the infra-red fixed point values (this would imply, among other things, the vanishing o f the quark m i x i n g ) , scale d e p e n d e n c e o f those p a r a m eters is no longer negligible in the presence of large Yukawa coupling (s) o f the third generation quarks. There are at least two obvious reasons for investigating this scale dependence, both related to the p r o b l e m o f u n d e r s t a n d i n g the pattern o f fermion masses. Firstly, given the experimental values o f the physical p a r a m e t e r s at the scale Mw, it is i m p o r t a n t to know their "initial" values at the, presumably much On leave of absence from Institute for Theoretical Physics, Warsaw University, ul. Ho~,a69, PL-00-681 Warsaw, Poland. 388
higher, scale where the structure o f the Yukawa coupling matrices is d e t e r m i n e d by, as yet unknown, physics. Secondly, in search for this new physics principles, there have been m a n y phenomenological attempts to relate the quark mixing p a r a m e t e r s to the values of the quark masses. It is an almost obvious necessary physical requirement that any such p a r a m etrization, to be considered as the one which m a y indeed reflect some underlying physics at some higher energy scale, should be stable against the R G evolution up to that scale. Therefore, it is interesting to c o m p a r e the scale dependence o f the elements I Vu[ with the scale dependence of various ratios hi~hi o f the eigenvalues o f Yukawa coupling matrices. Scale dependence is usually studied by means o f R G equations and the one-loop R G equations for the Yukawa matrices are well known in a large class o f theories [3]. However, to study the scale dependence of the observables I V,;[ and mi/rnj it is convenient ( a n d for several purposes even necessary) to have the R G equations for those observables in analytic form. In this paper we derive the R G equations for the mixing elements I V0[ and for the ratios h~/hj o f the eigenvalues of Yukawa coupling matrices. This is
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
Volume 257, number 3,4
PHYSICS LETTERS B
possible by use of the recently proposed [4 ] weak basis invariants directly related to the elements I V,jl. Next, we study the scale dependence of those quantities up to the unification scale in the presence of large Yukawa coupling(s) for the third generation quarks. Of course, the scale dependence of those physical parameters depends on the theory considered. After developing the general formalism we focus on the minimal standard model (MSM) and on the minimal SUSY model (SUSY). In this latter case we consider the possibility h~>> hb (and consequently v~~ v2) and also h~~ h b (b'2 > > / h ). W e observe interesting qualitative features of the scale dependence of various parameters and comment on their possible relevance for understanding the pattern of quark masses. (Our discussion for the SUSY model remains valid with only minor modifications for the general twoHiggs doublet models. ) The subject of the scale dependence of the quark mixing parameters has already been studied in the past by several authors [5-7]. In particular in ref. [6] the RG equations are given in analytic form for the elements [ V~j[ in the standard model and in ref. [7] the scale dependence of the mixing angles has been studied (by diagonalizing numerically the Yukawa coupling matrices at different scales) also in two-Higgs doublet models and in SUSY models in the presence of one large Yukawa coupling. We generalize those results in several directions. First, we derive the one-loop RG equations for the mixing elements I V,I in a large class of theories, including the cases studied earlier numerically. Thus, a qualitative insight into the behavior of different mixing angles and of the CP violation measure Joe in various models is obtained, our equations allow also for an easy study of models with two (three) large Yukawa couplings (top and bottom (and tau) couplings). Second, we also derive RG equations for the ratios of the eigenvalues of the Yukawa coupling matrices and compare them at the analytic level with the equations for I Vgjl. This may have interesting implications for models of the quark mass generation, which we briefly discuss. Let us consider the one-loop RG equations for the Yukawa matrices Y4, (A = U, D), in the following general form: Y~ = c,4 ~ + ~ aA~HB YA,
( 1)
28 March 1991
where H,~ = Y4 Y,+ is a hermitian matrix and t = ( 1/ 16n 2) ln(Mx//~). Eq. ( 1 ) includes the three specific cases we consider in this paper: ( 1 ) MSM with one Higgs doublet: CA "= G.4 -
T,
T = 3 Tr Hu + 3 Tr HD + T r H v , GU
Q 2.d_9 2_1_17 2 = o g 3 - - ~ g 2 - - ~-0gt ,
G D = 8 g 3 2 "t-~g2 9 2 "1- ~I g l2 , 3
auu=aDD=--5,
3 aUD-----aDU=~.
(2) Two-Higgs doublet model with one doublet giving mass to the up quarks and the other one to the down quarks and leptons: cA = G A - G ,
Tu=3TrHu,
TD=3TrHD+TrHE,
Gu = 8g 2 + 9g22+ ~2og~, 9 2
1 2
GD = 8g~ +~g2 + ~g~ , auu
=aDD
3 = - - 2,
aUD=aDU=--~
J
.
(3) SUSY model: c4 =GA -- TA,
TU = 3 T r H U , G
TD=3TrHD+TrHE,
v +~g2, u = y16g 32 --v3g5
GD=!~g~ + 3g 2 +~g17 2 , auu=aDD=--3,
aUD=aDU
=-1
,
where HE = YEY~- and YE is the Yukawa matrix for charged leptons. We notice an important difference in the coefficients ofeq. ( 1 ) between the MSM and the other two models: det ( a AB ) -~ 0 for MSM and det (aA t~ ) ~= 0 o t h e r w i s e . It implies different relations among the couplings in the IR fixed point. The MSM is special in that it allows in this point for the mass hierarchy among generations Hu-HDZC'I (in the approximations when the running of gauge couplings is neglected) whereas in the other cases Hu zc HD 3C~ at the fixed point. Of course, the above conditions are not satisfied in neither case but the approach to the fixed point is different in the two cases. As we shall see, this results in a qualitatively different scale dependence 389
Volume 257, number 3,4
PHYSICS LETTERS B
of various quantities in the MSM and in two-doublet models (including SUSY). From eq. ( 1 ) one can derive R G equations for the elements ]Vv[ and for the ratios hJhj. To this end we use the recently proposed [4 ] set of weak basis invariants l(n, m):
Tr([H~,H~]+[H~j,H'~]) l(n, m ) = Tr(H2, ) Tr(H~ )
(2)
The invariants (2) are directly related to the moduli squared of the mixing elements. For instance lira I(pN, q N ) = 2 ] V~h ]2(1 - ] V~, ] 2) ,
N~ow
(3)
where p, q = +_ 1, a = 2 + p , b - 2 + q . The remaining ] Vu[ 2 can be obtained from the unitarity condition. For the four elements IV, t,] in eq. (3) we have therefore
d
dtln
[Nab[ 2
d lim lnl(pN, qN), (4) -- 1_21V~b12dtx~o~
28 March 1991
sure Jc.e can be obtained from (5) using the relation
[8]
4JZ,e=41Vikl2[ Vtl[ 2 ] Vjk [2 I Viii 2 - ( 1 - I vikl 2 - I v, I 2 - I~kl 2 - [V~tl 2 + [ V~I21 ~ z l 2 + I V. 121 ~k12) 2 ,
where i¢j, k ¢ l . For the ratios of the eigenvalues of the Yukawa coupling matrices we get ~42 d
In ~
e`j
----aDD(h 2, - h 2 ) 3
+aDu ~ h~k(lVk, i2--1V~,i2), k--1
d lnhuJ 2 2 -dth..u~ = a u u ( h u , - h u ~ )
1--[Vab[ 2
3
+auD Z h~k(lVikl2--lVjkl2) k=l
and using eq. (1), after some calculation, one gets the following R G equations ~"
d~ln l
i=l ~" 1
h2i( ((~ih- I Vail2)
2 h 2t,
+ G~,~--~ i j~. h2p_h2f +2qaDu i=l ~
"
(8)
In the presence of large third generation Yukawa couplings, eqs. (5) give to a very good approximation the following: d
-d-t l n [Vi, I = - - a D u h t 2 - - a u D h b2,
]
( i j ) = (13), (31), (23), ( 3 2 ) ,
h2ui(((~ai-[V~b[2)
1
(7)
d - - I n [ VI2 [ = --aDuh2c--auDh2s dt
)
_aouh2 [V31 [2__ [ V1312 ]VI2[ 2
where, as before, p, q = + 1, a = 2 + p and b = 2 + q ; he,, is the i-eigenvalue of the matrix YD (and similarly, h~, of the matrix Yu) and
_auDh 2 [ G3 ]2_ 1V31 [2
R,:,,.t~j= Re ( V~, 17,, if, z, V#)
IVy, [2
= I Vat,[2[ Vq I 2+ [ V,v [21Vm]2+ ( 1 - ~5~i)( 1 - 6,',a) X ( 1 - I V a t , I 2 - I V , jI2-IVo, I 2 - [ V v l 2 ) .
d ~ t l n IV21 I = --aDuh2--auDh 2
(6)
d - - In IJcpl = --2aDuh 2 --2aUD h2. dt
(9)
As mentioned earlier, the RG equations for the remaining [Vql follow from (5) and the unitarity relations. The R G equation for the CP violation mea-
The accuracy of this approximation is very good, the differences between the numerical solutions of eqs. (5) and (9) are below 0.5% for all the cases consid-
~ Eq. (5) is the exact one-loop RG equation for IVc,b]2 if hd, < hd~< hd3 and similarly for the up sector.
~2 In the case of the minimal standard model eqs. (5) and (8) agree with the equations given in ref. [6].
390
Volume 257, number 3,4
PHYSICS LETTERS B
ered. In t h e s a m e a p p r o x i m a t i o n , f o r t h e r a t i o s hi/h I we get din
hal, 2 hb = --aDDhb--aDuh?'
d In ~h,, d--I
=
d In ha dt ~
----
i = 1, 2 ,
--auu h2 --aUD h2,
i = 1, 2 ,
2 _ a D u h 2 - - a D v h ~ ] J/32 --aDDhs
(10)
T h e a n a l y s i s o f eqs. ( 9 ) a n d ( 1 0 ) l e a d s us to t h e f o l l o w i n g c o n c l u s i o n s . In all t h e m o d e l s c o n s i d e r e d in t h i s p a p e r t h e scale d e p e n d e n c e o f t h e C a b i b b o a n gle ] G 2 ] is v e r y w e a k ( i t c h a n g e s b y less t h e n 0.1% between Mw and Mx) and qualitatively different from the behaviour of the small mixing elements. The
- f=lv,a[ - - - f=m,/mb
(a,/
......... f = m j m t
/
/
l a t t e r s h o w s t r o n g scale d e p e n d e n c e i n t h e p r e s e n c e o f large Y u k a w a c o u p l i n g s a n d are very s i m i l a r to e a c h o t h e r for all s m a l l e l e m e n t s . So t h e r a t i o s like I Vl3l / I V23l, etc., r e m a i n a l m o s t c o n s t a n t d u r i n g t h e e v o l u t i o n . H o w e v e r , in t h e M S M t h e s m a l l e l e m e n t s bec o m e l a r g e r w i t h i n c r e a s i n g e n e r g y w h e r e a s in S U S Y ( a n d in t h e g e n e r a l t w o - d o u b l e t c a s e ) t h e y b e c o m e s m a l l e r % T h u s , in M S M t h e n e a r e s t n e i g h b o u r m i x -
]2
d l n h~, = - a u u h2 --aUD h2 - - a u D h S~ I V23 I ,- . dt
',~ 15
28 March 1991
~3 All off-diagonal mixing angles vanish at IR fixed points both in the MSM and in two-doublet models. Thus, one could expect that they should decrease with decreasing energy. The opposite behavior of small mixing angles in two-doublet models is caused by the fact that the observed mass hierarchy among generations (which, according to eqs. (9), is crucial in determining the running of the mixing angles ) is allowed at the fixed point in the MSM and not allowed in two-doublet models (see the discussion before eq. (2)). The fact that the two-doublet models with h, ~ h b might approach fixed point behaviour for both quarks of the third generation does not change this behaviour. 10
(b)
~09 ~" 08 0.7
J r
1.0
E. . . . . . . . . . . .
06
..
-
-
[
--
-
0.5
....
0.5
80
120
,
160
200 240 mtop[GeV]
0.4
80
,
,
,
Iv,a]
f=rns/mb f
,
I00
,
rne/ri1
,
,
t
i
,
120
,
,
,
140
,
,
,
,
,
,
,
160 180 mtop[GeV]
09 08 07 0.6
1 0.5
"~',
~
/
- -
,~
-
-
......
80
~,, 9,
r=/vt=l f=m,/mb f = r n . / m t
100
120
i'
140
160 180 mtop[GeV]
Fig. 1. The ratios of ] V]3], ms~roband me~mr at Mx to their values at Mw as a function of the top quark mass for three models: (a) minimal standard model, (b) SUSY with t,2/v~= 2, (c) SUSY with ht=h b. Mx is the unification scale calculated separately for each model from low energy values of the gauge coupling constants. In the cases (a) and (b) the curve for I G3] overlaps almost the one for ms/mb. The Yukawa couplings ht and hb are evolved according to the RG equations for the traces of the matrices HA, obtained from eq. (1), neglecting light quark contributions. 391
Volume 257, number 3,4
PHYSICS LETTERS B
ing at large (unification?) scale m a y be similar for (12) and ( 2 3 ) families and the observed hierarchy I V231/I V~21 can be partially explained by the evolution in the presence o f a heavy top. In SUSY only Cabibbo mixing remains large at large scale and other mixing elements b e c o m e significantly smaller than at the scale Mw. This b e h a v i o u r o f small angles makes the C a b i b b o angle even m o r e exceptional in twoHiggs doublet models than in the MSM. In the presence o f a heavy top (with a mass close to the IR fixed point value) the scale d e p e n d e n c e o f small angles is a few per cent effect between 1 GeV and 100 GeV. The renormalization of small angles between Mw and the unification scale M x depends strongly on the value o f rn~ a n d on the model considered. Some numerical results are presented in fig. 1. In all the models the CP violation p a r a m e t e r Jcp behaves as a p r o d u c t o f two small mixing elements. The results given in fig. 1a a n d l b are essentially equivalent to those o b t a i n e d in ref. [ 7 ] by a different method. Turning now to a c o m p a r i s o n o f the scale dependence o f the elements I V,~I and o f the ratios h~/hj, we firstly observe that eq. ( 5 ) is different from eq. ( 8 ) . Therefore, the R G evolution does not seem to be naturally c o m p a t i b l e with the possibility that there are exact relations between the mixing elements and the quark masses. However, at the level o f the approxim a t i o n ( 9 ) and (10) there are certain regularities. The scale d e p e n d e n c e o f the C a b i b b o angle is similar to the behaviour o f the ratios hd/hs and hu/hc but only numerically. This d e p e n d e n c e is so weak that we d i d not plot it in fig. 1. F o r small elements I V,j l there is quite clearly a resemblance to the b e h a v i o u r of the h~/hj. In the M S M and in two-doublet models (including S U S Y ) with ht >> ho they run as the ratios hdi/hb, i = 1, 2. On the other hand, in the latter models but with ht'~hb (for v2/Vl >> 1) the angles run as x/~o,/hb ( a n d also as x/h~,/ht ). Thus, the R G evolution is c o m p a t i b l e with some a p p r o x i m a t e relation between the mixing elements (in particular small ele-
392
28 March 1991
m e n t s ) and the ratios o f the quark masses. As a final remark we note that the IR fixed p o i n t value o f the top quark mass in the M S M is m t ~ 230 GeV. In the two-doublet m o d e l it changes from m, ~ 180 GeV for ht >> hb and v2 -~ v~ to mt ~ 240 G e V for ht ~ hb ~4. In the SUSY m o d e l the IR fixed point top mass is smaller: mt ~ 150 GeV for ht >> hb ( 1:2~ vl ) and m~ ~ 190 GeV for h~ ~ h b. One of us (S.P.) would like to acknowledge the very kind hospitality o f T . N . Troung in the Centre de Physique Th6orique, l~cole Polytechnique, Palaiseau, where part o f this work was done. This work is partially s u p p o r t e d by the Polish research project CPBP 01.03. ~4 See also ref. [9].
References [ 1] B. Pendleton and G.G. Ross, Phys. Lett. B 98 ( 1981 ) 291; C.T. Hill, Phys. Rev. D 24 ( 1981 ) 691; E.A. Paschos, Z. Phys. C 26 (1984) 235. [2] J. Kubo, K. Sibold and W. Zimmermann, Phys. Lett. B 220 (1989) 191;Nucl. Phys. B 259 (1985) 331; W.J. Marciano, Phys. Rev. Lett. 62 (1989) 2793; W.A. Bardeen, C.T. Hill and M, Lindner, Phys. Rev. D 41 (1990) 1647. [3] T.P. Cheng, E. Eichten and L.F. Li, Phys. Rev. D 9 (1974) 2259. M. Machacek and M. Voughn, Nucl. Phys. B 236 (1984) 221. [4] M. Olechowski and S. Pokorski, P;,ys. Lett. B 231 (1989) 165. [5] E. Ma and S. Pakvasa, Phys. ~,:~. ? ~6 (1979) 43; Phys. Rev. D20 (1979) 2899. [6] K. Sasaki, Z. Phys. C 32 (1986) 149. [7] B. Grz~dkowski and M. Lindner, Phys. Lett. B 193 (1987) 71; B. Grz~tdkowski, M. Lindner and S. Theisen, Phys. Lett. B 198 (1987) 64. [8] C. Hamzaoui, Phys. Rev. Lett. 61 (1988) 35. [9] C.D. Froggatt, I.D. Knowles and R.G. Moorhouse, Glasgow University preprint GUTPA/90/6-1 (1990).